##########################################################################
#  Copyright (C) 2006 Jaap Spies, jaapspies@gmail.com
#  Copyright (C) 2006 William Stein, wstein@gmail.com
#
#  Distributed under the terms of the GNU General Public License (GPL):
#
#                  http://www.gnu.org/licenses/
##########################################################################

def A079912(n):
    r"""
    function returns solutions to the Dancing School problem with 7 girls

    The value is $per(B)$, the permanent of the (0,1)-matrix $B$
    of size $7 \times 7+n$ with $b(i,j)=1$ if and only if $i \le j \le i+n$.

    We use precalculated values for $n = 0, ..., 4$ For $n > 4$ we use
    the polynomial $n^7-14*n^6+126*n^5-700*n^4+2625*n^3-6342*n^2+9072*n-5840$ 
    calculated with the function dance.

    See http://www.jaapspies.nl/mathfiles/dancing.sage

    See Nieuw Archief voor Wiskunde 5/7 nr. 4 December 2006 p. 285

    INPUT:
        n -- non negative number

    OUTPUT
        integer

    EXAMPLES:
        sage: A079912(4)
        5794

        sage: A079912(10)
        3675680

        sage: A079912(20)
        693838800

        sage: A079912(1000)
        986125302618667066160


    AUTHOR:
        - Jaap Spies (2006-12-10)
    """
    b = [1, 8, 133, 1044, 5794 ] 

    if n >= 0 and n <= 4:
        return b[n]
    else:
        return n^7-14*n^6+126*n^5-700*n^4+2625*n^3-6342*n^2+9072*n-5840

def A079912_list(n):
    r"""
    function returns list of solutions to the Dancing School problem with 7 girls

    We use precalculated values for $n = 0, 1, 2, 3, 4$. For $n > 4$ we use the
    polynomial $h^7-14*h^6+126*h^5-700*h^4+2625*h^3-6342*h^2+9072*h-5840$ 
    calculated with the function dance.
    See http://www.jaapspies.nl/mathfiles/dancing.sage

    INPUT:
        n -- non negative number

    OUTPUT
        list of integers

    EXAMPLES:
        sage: print A079912_list(10)
        [1, 8, 133, 1044, 5794, 24720, 86608, 260720, 693552, 1666000, 3675680]      

        sage: print A079912_list(1)
        [1, 8]

        sage: print A079912_list(0)
        [1]

        sage: A079912_list(3)
        [1, 8, 133, 1044]


    AUTHOR:
        - Jaap Spies (2006-12-10)
    """

    h = QQ['h'].gen()

    b = [1, 8, 133, 1044, 5794 ] 

    if n >= 0 and n <= 4:
        return b[:n+1]
    else:
        pol =  h^7-14*h^6+126*h^5-700*h^4+2625*h^3-6342*h^2+9072*h-5840  
        a = [pol(i) for i in range(5, n+1)]
        for i in range(5):
            a.insert(i,b[i])
        return a